1. Introduction to the phenomenology of HiTc superconductors.
Patrick Lee and T. Senthil
MIT

2. Introduction to experimental methods.
Basic physics: doped Mott insulator. (Early sections in Lee, Nagaosa and Wen, Rev Mod Phys,78,17(2006) and Lee, Reports of Progress in Physics, 71,012501(2008))
Introduction to experimental methods.
Thermodynamic measurements: specific heat, spin susceptibility.
Transport: resistivity, Hall, magneto-resistance (angle dependence ADMR).
thermo-power, thermal conductivity.
quantum oscillations.
AC conductivity, optical, microwave and IR, time domain spectroscopy.
Neutron scattering
NMR
ARPES
Tunneling and STM.
3. Pseudo-gap physics.

4. Corner sharing octahedrals.eg
t2g
dxy,dyz,dzx
dz2, dx2-y2
Octahedral
field splitting
X2-y2 z2

5. CuO plane: strongly-correlated electron system
One hole per site: should be a metal according to band theory.
Mott insulator.
Undoped CuO2 plane:
Mott Insulator due to
e- - e- interaction
Virtual hopping induces
AF exchange J=4t2/U
CuO2 plane with doped holes:
La3+  Sr2+: La2-xSrxCuO4 t
Here is the illustration of the electron transport in the undoped CuO plane first. (Technion: you’ve probably had a more elaborate intro to this subject from Assa Auerbach)
Doping restores the ele conductivity by creating cites at which the holes can jump without paying the cost in Coulomb repuls. The doping is achieved by chemical subitution of La by Sr or by adding additional oxygen into the lattice. As a result of doping, the holed can now move around. However, as you see, doping destroys the AFM order very quickly because this spin after the hole move there wants to be down rather than up. The most famous example of a HTC material is LSCO, which is very similar to the originally discovered LBCO.
So, the 2 basic features which characterize the HTCS materials:
(i) key structural unit shared by all the materials is CuO plane. The interplane correlations are weak, so the behavior is quasi-2D
(ii)HTSC’s are created by doping Mott Insulators
This combination creates the fundamentally different behavior as compared to the “old”, regular superconductors like merculry, Al, NbSe2, etc.
First,CuO plane is a 2D spin-half Heisenberg AFM with very large coupling J, of 1500K. The AFM is due to the superexchange interaction between Cu spins through Oxygen p-orbitals.
Second, there is one hole on every Cu cite, so the conduction band is only half filled. However, hoping between cites is prohibited because the Coulomb energy of double-occupied cite is too large. Therefore, strong el-el correlation plays an important role in this system. This is in contrast to regular Fermi-liquid metals, where el kinetic energy is dominant. Mott Insul is a material, like LaCuO, in which the conductivity vanishes at decreasing T, even though the band theory predicts it to be metallic.
(J=t^2/U)
The combination of the 2-dim and proximity to Mott Insul leads to a new physics. The SC is only one of the welth of new phenomena observed and still poorly understood in HTSC.

6. Charge transfer insulator.
Mott insulator
Charge transfer insulator.
Hole picture
Electron picture
Ogata and Fukuyama, Rep. Progress in Physics, 71, (2008)

7. Also from Raman scattering.
By fitting the spin wave dispersion measured by neutron scattering. (also needs a small ring exchange term.)
Also from Raman scattering.
Spin flip breaks 6 bonds, costs 3J.
Largest J known among transition metal oxide, except for the Cu-O chain compound where J=220meV.

8. Doping a charge transfer insulator: The “Zhang-Rice singlet”
Due to AF exchange between Cu and O, the singlet symmetric orbital gains a large energy, of order 6 eV. This singlet orbital can hop with effective hopping t given by:
Symmetric orbital
centered on Cu.
Anti-symmetric orbital

9. What is unique about the cuprates?
Pure CuO2 plane
Single band Hubbard model, or its strong
coupling limit, the t-J model. 
Dope
holes t J
t  3 J
1) low dimension
2) H = J  Si · Sj nn
large J = 135 meV
Competition:
t favors delocalization of electrons
J favors ordering of localized spins
3) quantum spin S =1/2
(NNN hopping t’ may explain asymmetry
Between electron and hole doping )

10. Fermi liquid theory in a nut-shell:
1. Well defined quasi-particles exist provided 1/t<<E near the Fermi energy. Usually 1/t ~ T^2. The electron spectral function has the form
2. Luttinger Theorem: volume of Fermi surface is the same as free fermion, ie For n carriers it is n/2 mod 1 of the area of the BZ.
3. Physical quantities are given by free fermion expressions except for Landau parameters. Except for tunneling, z does not appear.

11. Doping x holes in a Mott insulator.?
Low doping: AF order. Unit cell is doubled. We have small pockets of total area equal to x times the area of BZ.
Large doping: no unit cell doubling.
Total Fermi surface area is
Area in the reduced BZ is

12. Single hole.
Small doping
Superconducting state.
Fermi liquid.
Pseudo-gap.

13. How many ways does Nature have to deal with doping a Mott insulator?
Electron doped.
AF with localized carriers.
Micro phase separation: stripes
3 Dimension. Brinkman-Rice Fermi liquid.
Organic ET salts.
Metal-insulator transition by tuning U/t.
Possibility of a “spin liquid”.
Doping yields a superconductor.
A second family of HiTc superconductors!

14. Electron doped side: AF persists to x=0
Electron doped side: AF persists to x=0.13 and the doped electrons are localized.
What is the origin of the p-h asymmetry? Hopping of electron on Cu (d10) is physically different from hopping of a Zhang-Rice singlet located on the oxygen. One possibility is polaron effect is stronger on the electron side.

15. J=31 meV
X<0.2 commensurate spin order, localized hole. (polaron effect?)
0.2<x diagonal stripe with 1 hole per Ni. (microscopic phase separation into Ni2+ and Ni3+).
Non-metallic until x=0.9
Smaller J means it is deeper in the Mott phase. Effective hopping is also small and polaron effects favor localized carriers.
Now ½ hole per linear distance along the stripe (2 Cu sites) : mobile charge.

16. Tokura et al, PRL 70, 2126 (1993).
3 dim perovskite structure.
X=0 is a band insulator, x=1 is a Mott insulator.
For x=1, Ti is d1 and has S=1/2. Very small optical gap (0.2eV). Surprisingly small TN=150K, (reduced due to orbital degeneracy).
Specific heat = gT

17. This is an example of “Brinkman-Rice Fermi liquid”.
Diverging mass near the Mott insulator. m*/m=1/xh, z=xh.
s= e^2nt/m* is proportional to xh , even though Fermi surface is “large” and has volume x=1-xh as inferred from the Hall effect.

18. Metal- insulator transition by tuning U/t.
AF Mott insulator
Cuprate superconductor
Tc=100K, t=.4eV, Tc/t=1/40.
Organic superconductor
Tc=12K, t=.05eV, Tc/t=1/40.
metal x

19. anisotropic triangular lattice t’ / t = 0.5 ~ 1.1
Q2D organics k-(ET)2X ET
dimer model X t’ t
Mott insulator
X = Cu(NCS)2, Cu[N(CN)2]Br, Cu2(CN)3…..
anisotropic triangular lattice
t’ / t = 0.5 ~ 1.1

20. Q2D antiferromagnet k-Cu[N(CN)2]Cl t’/t=0.75
Is the Mott insulator necessarily an AF?
“Slater vs Mott”.
Until recently, the experimental answer is yes.
A digression on spin liquid.
Q2D antiferromagnet
k-Cu[N(CN)2]Cl
t’/t=0.75

21. t’/t=1.06 Q2D spin liquid Q2D antiferromagnet k-Cu2(CN)3
k-Cu[N(CN)2]Cl
t’/t=1.06
No AF order down to 35mK.
J=250K.
t’/t=0.75

22. Magnetic susceptibility, Knight shift, and 1/T1T
C nuclear
[A. Kawamoto et al. PRB 70, (04)]
Finite susceptibility and 1/T1T at T~0K : abundant low energy spin excitation (spinon Fermi surface ?)

23. From S. Yamashita,.. K. Kanoda, Nature Physics, 4,459(2008)
g is about 15 mJ/K^2mole
Something happens around 6K.
Partial gapping of spinon Fermi surface due to spinon pairing?
Wilson ratio is approx. one at T=0.

24. More examples have recently been reported.

26. ET2Cu2(CN)3 Insulator spin liquid ET2Cu(NCS)2 9K sperconductor
Thermal conductivity
ET2Cu2(CN)3 Insulator spin liquid
ET2Cu(NCS)2 9K sperconductor
Belin, Behnia, PRL81,4728(1998)
M. Yamashita ...Matsuda ,Nature Physics 5,44(2009)

27. Doping of an organic Mott insulator.
Superconductivity in doped ET, (ET)4Hg2.89Br8, was first discovered Lyubovskaya et al in Pressure data form Taniguchi et al, J. Phys soc Japan, 76, (2007).

28. Note the common feature of high Tc and organics:
Proximity to Mott insulator.
singlet and d-wave pairing.
Is it possible to have superconductivity in purely repulsive models, and if so, how do we understand it?
Note that in d-wave pairing, we avoid on-site repulsive energy. By making singlet pairs, we can gain exchange energy.

29. 1. The one hole problem.
Theory for t-J model: self consistent Born approx. of hole scattered by AF magnon works very well. (Kane,Lee and Read, 1989 , Schmitt-Rink et al,….).
Main conclusions: the dispersion is given by an effective hopping of order J. The hole spectrum has a coherent part with relative spectral weight (J/t) and a broad incoherent part spreading over t.
ARPES data: review by X. J. Zhou et al, cond-mat )
Not the whole story: line width very broad (300meV) and comparable to dispersion. To explain this, need to include strong electron phonon coupling (polaron). Line-shape is interpreted as Franck-Condon effect as in molecular H2. However, the peak of the spectral function is still given by the bare dispersion.
Message: one band t-J model works, but need strong e-phonon coupling.

30. Ideal for 2 dim. Assume parallel momentum is conserved
Ideal for 2 dim. Assume parallel momentum is conserved. Measure ejected electron energy and infer the energy and momentum of the hole left behind.
Surface sensitive probe.
Resolution a few meV.
Recent Laser ARPES employs VUV lasers (about 7 eV). Energy resolution 0.26meV. Deeper penetration. Limited k space coverage. No tunability.

31. VUV Laser Synchrotron 0.26 5~15 0.0036 (6.994eV) 0.0091 (21.1eV)
Advantages and Disadvantages of VUV Laser ARPES
Light Source
VUV Laser
Synchrotron
Energy Resolution (meV)
0.26
5~15
Momentum Resolution (Å-1)
0.0036
(6.994eV)
0.0091
(21.1eV)
Photon Flux(Photons/s)
1014~1015
Electron Escape Depth (A)
30~100
5~10
Photon Energy Tunability
Limited
Tunable
k-Space Coverage
Small
Large
Laser and Synchrotron are complementary. 31

32. Bi2Sr2CaCu2O8+d Bi-2201 (Bi2Sr2CuO6+x) BSCCO or Bi-2212 LSCO YBCO
Cleavage plane. Disorder. .
Simple, x is known, disorder. Low Tc.
Cleanest. Doping by varying oxygen conc. on chains.

33. Eisaki et al, PRB 69, (2004)
With further increase of layers, Tc does not go up further. The inner planes have less hole and may be AF ordered.

34. 2. Small doping. DC transport. Boltzmann conductivity: s=ne^2t/m
Ando et al, PRL 87, (01)
Hall effect:
RH=1/nec
Anomalous T dependence.
x=0.03 sample, from Padilla et al, PRB72,060511(2005)

35. Optical conductivity Timusk and Statt,Rep Prog Phys 62,61 (99)
From reflectivity or ellipsometry, deduce Re and Im parts of s(w).
Drude formula for simple metal:
Extended Drude formula:
Include high frequency incoherent part.
Padilla et al, PRB72,060511(05)

36. Conclusion from transport measurements:
No divergent mass enhancement. m*/me~4.
Drude spectral wt (n/m*) is proportional to x with no T dependence. This wt becomes the superfluid density in the SC.
Scattering rate is roughly 2kT and becomes linear in w at high frequencies.
Weight of delta function is the superfluid density and is proportional to x

37. Neutron scattering:
If there is long range AF order, Bragg peaks appear at G’s.
The direction of the ordered moment can be determined by rotating G.
In the absence of long range order, we can measure equal time correlation function by integrating over w.

38. Local moment picture works
Local moment picture works. Reduced from classical moment of unity due to quantum fluctuations of S=1/2.

39. NMR
Local probe. Does not require large samples. Very important for the study of new materials.
1. Knight shift. Proportional to spin susceptibility, but free from impurity contributions. Line is often broadened by random distribution of local fields. Need good quality material. The shift and onset of line broadening can be used to measure spin order.
c=cS (T)+ cVV +ccore+cimpurity(T)
K=KS (T) + KVV + Kcore
KS ~ cS
KVV ~ cVV
2. Spin relaxation rate. Measures the low energy spectrum of spin fluctuations.
Form factor F(q) peaks at different q for different sites.
For example, planar oxygen site does not see AF q=(p,p), but Cu site does.
For metals, Korringa relation:
3. NQR. Measure local electric field distribution.

40. Knight shift on different sites have identical T dependence.
One component vs two component system: validity of the one band Hubbard model.
Knight shift on different sites have identical T dependence.
Takigawa et al PRB43, 247 (91).

41. Theoretically, C. Varma believes that 3 band Hubbard model with interaction V between Cu and O is needed. He proposes the existence of orbital currents in the plane between Cu and O. These currents occur within the unit cell and does not change the unit cell.
Orbital currents have been observed by neutron scattering. The onset of these currents seem related to T*, the pseudo-gap scale.
However, the moments are about 45 degrees from the plane. Numerical studies find orbital currents between planar and apical oxygen. (Weber et al, ArXiv ). Perhaps these effects do not affect the Fermi surface.
There is also reports of T breaking (ferromagnetic like) by polar Kerr effect at slightly lower temperature. (Xia,…Kapitulnik,PRL 100, (08))
Li ..Bourges, Greven, Nature 455, 372 (2008).

42. 3. Properties of the superconductor.
Pairing is d symmetry.
Phase sensitive measurements.
tri-crystal experiment, IBM ½ flux vortex at the junction. Standard hc/2e votex everywhere else.
2. Corner SQUID.
Wollman et al 1993.
Tsuei and Kirtley Rev Mod Phys 2000.

43. ARPES. Node along diagonal.
Dirac cone characterized by vF and vD.
Ding et al Nature 382,

44. Importance of phase fluctuations.
Superfluid stiffness Ks is related to the Drude spectral wt..
It is measured by London penetration depth.
Uemura plot: linear relation between Tc and ns/m*.
Microwave cavity perturbation expt, or by muon precession relaxation rate which measures the magnetic field distribution near the vortex. Note very long l (several thousand angstrom) implies very small stiffness or superfluid density.
Thermal excitation of nodal qp gives linear T reduction.
From Boyce et al, Physica C 341,561 (00)

45. +ve Muons relax to certain (often unknown) sites.
A distribution of magnetic field (eg caused by the overlapping fields of vortices) causes a damping of the oscillations.
Another set-up is zero field muSR, which is very sensitive to static (on the scale of the muon lifetime of 2 micro-sec) internal magnetic field (as low as a few gauss) due to magnetic ordering or spin glass freezing.

46. In 2D phase fluctuations destroy SC order via the Kosterlitz-Thouless mechanism of proliferation of vortices and anti-vortices. They predict a universal relation:
Then Tc is controlled by Ks, not by the energy gap as in BCS theory. Strong violation of BCS relation 2D/kTc~4.
The dynamics of phase fluctuations is probed by microwave conductivity by Corson et al Nature 398,221 (1999) in UD Bi-2212 Tc 74K.
For a SC:
(W=1/t)
Scaling function:
(More about fluctuation SC via Nernst effect and diamagnetism later.)

47. m*/m=1+l, but l usually has no isotope effect.
YBCO
Summary:
Substantial isotope effect on Tc for underdoped, but little or no isotope effect for optimal and overdoped.
However, there is isotope effect on ns/m* for all doping. (unexplained: needs better understanding of e-phonon in strongly correlated materials.)
On the other hand there is no isotope effect on Fermi velocity by Laser ARPES, while there is shift in “kink” energy. (Iwasawa..Dessau,PRL101,157005(08)
m*/m=1+l, but l usually has no isotope effect.
Khasanov …Keller, PRB 73, (06)
Qualitatively consistent with the idea that in UD, Tc is controlled by ns/m*.
If Tc~ns/m*, we expect DTc/Tc=-2Dl/l (line A), but data is closer to DTc/Tc=-dl/l. (line B)
However, in practice Tc has a more complicated dependence on ns/m*.

48. Other probes of nodal quasi-particles:
1. Quasi-particle dispersion shifted by electromagnetic gauge field A.
Volovik (1993) pointed out that near a vortex,
Set R to the average spacing between vortices.
Predicts a specific heat which goes as sqrt(B) and observed by K. Moler.
2. Universal ac conductivity and thermal conductivity. ( Lee, 93, Durst and Lee 2000)
Use to measure vD/vF.
Taillefer, PRL 79, 483 (97)

49. Raman scattering (electronic).
Devereaux and Hackl, Rev Mod Phys 79, 175(2007)
Non-resonant
resonant

50. Probe particle-hole charge excitation with a form factor.
Expand the polarization tensor in terms of irreducible representation of lattice point symmetry. For square lattice:
g(k)=
g(k)=

51. Expected contribution from quasi-particle, quasi-hole excitation.
The initial slope is proportional to t.
The broad continuum comes from incoherent electronic excitations.

52. Summary:
The superconducting state is singlet d-wave pairing. The nodes dominate low temperature properties and are well characterized.
In the underdoped region, Tc is determined by phase fluctuation and not by the vanishing of the pairing gap. As a result, the energy gap is large even though Tc is small.
While unusual, a lot of the physical properties of the superconducting state at low temperature can be understood based on a conventional physical picture.
As we will see, questions remain as to what happens at higher temperature above Tc and in a high magnetic field which restores the resistive state. Furthermore, the precise behavior of the gap near the anti-node (0,p) remains to be clarified.